Phys. Status Solidi B 246, No. 11–12, 2744–2749 (2009) / DOI 10.1002/pssb.200982301 p s sb

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basic solid state physics

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Single-wall carbon nanotubes:

spintronics in the Luttinger liquid phase

Balazs Dora*,1, Miklos Gulacsi1, Adam Rusznyak2, Janos Koltai2, Viktor Zolyomi3,4, Jeno Kurti2,and Ferenc Simon5

1 Max-Planck-Institut fur Physik Komplexer Systeme, Nothnitzer Str. 38, 01187 Dresden, Germany2 Department of Biological Physics, Eotvos University, Pazmany Peter setany 1/A, 1117 Budapest, Hungary3 Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK4 Research Institute for Solid State Physics and Optics of the Hungarian Academy of Sciences, Konkoly-Thege M. ut 29-33, 1121

Budapest, Hungary5 Budapest University of Technology and Economics, Institute of Physics and Solids in Magnetic Fields Research Group, Hungarian

Academy of Sciences, P.O. Box 91, 1521 Budapest, Hungary

Received 30 April 2009, revised 11 May 2009, accepted 18 August 2009

Published online 13 October 2009

PACS 71.10.Pm, 73.63.Fg, 76.30.–v

* Corresponding author: e-mail dora@pks.mpg.de, Phone: þ49 351 871-2417, Fax: þ49 351 871-1999

We study the electron spin resonance (ESR) intensity in the

Luttinger liquid phase of carbon nanotubes. The ESR

measurables such as the signal intensity and the line-width

are calculated in the framework of Luttinger liquid theory with

broken spin rotational symmetry as a function of magnetic field

and temperature. The linewidth is broadened significantly at

high temperatures, and the intensity is well describes by

Lorentzians. At very low temperatures, however, the ESR

lineshape becomes asymmetric around the resonance, and is

characterized by threshold behavior. These observables are of

special importance for the spintronics applications of carbon

nanotubes.

� 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction The effect of electron–electron inter-action is an intensively investigated subject of condensedmatter physics. Low dimensionality tends to further enhancethe importance of correlation, driving the systems intopeculiar low temperatures phases. One-dimensional inter-acting electrons usually form a strange metallic phase,known as Luttinger liquid. In such a state of matter, thefermionic quasiparticle picture breaks down, and the lowenergy excitations are characterized by a set of bosoniccollective density modes (phonons) of the system, which arethe true elementary low-energy excitations. A rather faithfulrealization of this phase is expected to occur in carbonnanotubes, which is confirmed by both the theoretical [1–5]and experimental [6–9] side. In general, low-dimensionalcarbonaceous systems, fullerenes, carbon nanotubes, andgraphene exhibit a rich variety of such phenomena includingsuperconductivity in alkali doped fullerenes [10], quantizedtransport in single-wall carbon nanotubes (SWCNTs), andmassless Dirac quasi-particles showing a half integerquantum hall-effect in graphene even at roomtemperature [11].

Due to the enhanced effect of correlations in low-dimensional SWCNTs, the physical observables areexpected to deviate significantly from those in a conven-tional Fermi liquid phase. The applicability of SWCNTs forspintronics can only be decided after a careful analysis oftheir spin relaxation properties. Extensive research in thisfield is motivated by the orders of magnitude longerconservation of the electron spin alignment in metals ascompared to their momentum conservation time [12]. Anideal tool to study the spin relaxation properties of metals isthe electron spin resonance (ESR) technique. The spindegeneracy of the energy levels is broken by the applicationof magnetic field. In the presence of an additional smalltransversal microwave magnetic field, resonant absorption isinduced between the split levels. By changing the value ofthe longitudinal magnetic field, one determines the positionand broadening of the resonance, thus leading to the spinlifetime (the inverse of the linewidth) and g-factor (theposition of the resonance). In three-dimensional metals,the ESR signal intensity is proportional to the Pauli spin-susceptibility, the ESR line-width and g-factor are

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Phys. Status Solidi B 246, No. 11–12 (2009) 2745

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determined by the mixing of spin up and down states due tospin-orbit (SO) coupling in the conduction band. These ESRmeasurables are affected when correlations are present andthus their study holds information about the nature of thecorrelated state.

This motivated a decade long quest to find the ESR signalof itinerant electrons in SWCNTs and to characterizeits properties in the framework of the expectedcorrelations [13–16]. However, to our knowledge noconclusive evidence for this observation has been reported.Let us summerize briefly the current experimental situation:doped SWCNT does not belong to the Luttinger liquid (LL)universality class, but rather forms a Fermi liquid. In thiscase, the properties of the ESR signal is well documented[14]. In undoped SWCNT, which forms an LL according tophotoemission spectroscopy [8, 9], the picture is less clear,and the observation of various signals ranging from Paulithrough Curie to super-Curie explains the debate on theinherent signal of LL. An often cited argument for thisanomalous absence of the ESR signal is the largeheterogeneity of the system, the lack of crystallinity, andthe presence of magnetic catalyst particles [13–16].However, this would also prevent the observation of theESR signal of doped SWCNTs [14]. Thus the aboveproperties of SWCNTs should hinder the observation ofthe ESR signal also for the Fermi liquid state, which is clearlynot the case. We suggest that the LL state inherently prohibitsthe observation of ESR of the itinerant electrons, calling for arealistic description of such experiments. A recent exper-iment by Kuemmeth et al. [17] shed new light on the spindegree of freedom of SWCNTs. It was shown that SOcoupling and correspondingly the lifting of the spinrotational invariance is unexpectedly large. As we showbelow, this results in a uniquely large homogeneousbroadening of the ESR line which explains the absence ofan intrinsic ESR signal of SWCNTs. Previous theories ofESR in the SWCNTs concentrated on non-interacting spinsector with SO coupled quasiparticles [18, 19].

One-dimensional interacting fermions usually form aLuttinger liquid, which replaces the canonical Fermi liquidpicture in higher dimensions. The quasiparticle descriptionbreaks down and low energy properties are described bycritical phenomena of collective modes for this state ofthe matter due to restricted dimensionality and interactions[20, 21]. This results in anomalous power-law dependence ofcorrelation functions at low energies, with critical exponentschanging continuously with the interaction strength. Anotherhallmark of LL is spin-charge separation, spin and chargeexcitations propagate with different velocities due tothe complicated many body ground state, caused by theinteractions.

Here, we study the ESR signal in an LL with broken spinrotational symmetry. While at low temperatures thecharacteristic non-integer power laws characterize theresponse, the high temperature behavior crosses over tothe standard Lorentzians, whose width, in contrast to theFermi liquid picture [22], is determined by the LL

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parameters. We show that this explains the absence ofitinerant ESR in this system by combining DFT calculationsof the spin-susceptibility on metallic SWCNTs with a criticalevaluation of the experimental conditions.

To describe a metallic SWCNT, we apply an effectivelow-energy theory. We neglect the ‘‘flavour’’ index comingfrom the two K points [19] since we are interested in the spinproperties only. The standard LL Hamiltonian is expressedas a sum of independent spin and charge excitations as

H ¼Xn¼c;s

�hvn2

Zdx Kn P

2n þ

1

Kn@xfnð Þ2

� �, (1)

where Kn are the LL parameters, n¼ c, s denotes the chargeand spin sector, respectively, Pn and fn are canonicallyconjugate fields with velocity vn. The LL parameter inthe spin sector, Ks¼ 1 for SU(2) symmetric models as thesepreserve the spin rotational symmetry. However, thepresence of SO and magnetic dipole–dipole interactionbetween the conduction electrons [20, 23] produces spindependent interactions and breaks the spin rotationalsymmetry, leading to Ks 6¼ 1. In addition, these processesare also responsible for the g-factor anisotropy. The motionof the electrons is restricted in the x direction.

The original fermionic field operators are expressed interms of the bosons as

Crs xð Þ ¼ hrsffiffiffiffiffiffiffiffiffi2pa

p �

� exp i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip

2rfc xð Þ þ rsfs xð Þ þQc xð Þ þ sQs xð Þð Þ

r� �;

(2)

which are needed to express the spin operators in thebosonic language, hrs is the Klein factor, Qn xð Þ ¼�R x

�1 dx0Pn x0ð Þ, r¼R/L¼� denotes the chirality of theelectrons, and s¼� is the electron spin.

The ESR experiments are performed in a longitudinalstatic magnetic field, B, applying a transversal perturbingmicrowave radiation with a magnetic component, B?. Forthe ESR description, the above Hamiltonian is completedwith the Zeeman term:

HZ ¼ �gmBB

Zdx@xfs xð Þ: (3)

The ESR signal intensity is given by the absorbedmicrowave power that is [22]:

I vð Þ ¼ B2?v

2m0

x00? q ¼ 0;vð ÞV , (4)

where m0 is the permeability of the vacuum, x00? the

imaginary part of the retarded spin-susceptibility for thetransversal direction, and V is the sample volume. The spinoperators required to calculate x00

? are

S� xð Þ ¼Xr;r0

exp i r0 � rð ÞkFð ÞCþr� xð ÞC r0� xð Þ: (5)

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2746 B. Dora et al.: Single-wall carbon nanotubes: spintronics in the Luttinger liquid phasep

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Since ESR measures the q¼ 0 response, only the r ¼ r0

terms contribute, the others contain fast oscillating terms� exp �2ikFð Þ and average to zero.

The Zeeman term in Abelian bosonization is the simplestwhen the longitudinal magnetic field points in the spinquantization axis (the z-axis). For a different field orien-tation, the Zeeman term becomes more complicated but itcan be rotated along the z direction, at the expense ofchanging the LL parameters Kn [23]. The external magneticfield further lowers the SU(2) symmetry in addition to the SOand dipole–dipole interactions, resulting in a furtherrenormalization of Ks.

From now on, we set �h ¼ kB ¼ gmB ¼ 1 and they will bereinserted whenever necessary. The retarded spin-suscepti-bility is built up from correlators of the type [24]

� 20

Sþ x; tð ÞS� 0; 0ð Þh i ¼ c2?

pTa=vssin h pT x=vs � t þ iað Þ½ �

� �2þg

� pTa=vssin h pT x=vs þ t � iað Þ½ �

� �g

expibx

vs

� �,

(6)

where b¼KsB, c? is determined by the short distancebehavior and cannot be obtained by the methods used here.Here we introduced the g ¼ Ks þ 1=Ks � 2ð Þ=2 parameter,which encodes information about spin symmetry breakingprocesses. Upon Fourier transformation, we obtain theretarded spin-susceptibility. From a simple scaling analysis,we can conjecture the behavior of the retarded spin-susceptibility as

x q ¼ 0;vð Þ?� max B;v; T½ �2g , (7)

20

which is confirmed below.Putting all this together, we find for the ESR intensity

(following Ref. [21], Appendix C):

12

14

16

18

rb.units)

K s = 1 .1

K s = 1 .2

K s = 1 .3

I vð Þ ¼ �A sin pgð Þv 2paT

vs

� �2g

� Im F 2 þ g; kþð ÞF g; k�ð ÞF 2 þ g; k�ð ÞF g; kþð Þ½ �,(8)

10)(a

where

6

8I (B

K s = 1 .4

k� ¼ v� b

2pT, (9)

4

0 0.2 0.4 0.6 0.8 10

2

F x; yð Þ ¼ B

x� iy

2; 1 � x

� �, (10)

gµBB/ ω

Figure 1 (online colour at: www.pss-b.com) The ESR signalintensity, Eq. (11) is shown at T¼ 0 as function of the magneticfield for Ks¼ 1.1, 1.2, 1.3, and 1.4. The resonance occurs atgmBB=�hv ¼ 1=Ks. The asymmetric intensity reflects the LL natureof the ground state.

where B x; yð Þ ¼ G xð ÞG yð Þ=G xþ yð Þ is Euler’s beta func-tion, G(x) is Euler’s gamma function, A is a constant, whosevalue is determined further below. In the g ¼ 0 limit, SU(2)spin symmetry is conserved by the Hamiltonian and the ESRresonance becomes completely sharp, located at �B as� B2d v� Bð Þ.

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The influence of interactions is most clearly seen atT¼ 0, when the ESR signal is completely asymmetric aroundv ¼ �b, and cannot be approximated by Lorentzians:

I vð Þ ¼ Aa

2vs

� �2g

sin2 pgð ÞG2 1 � gð Þg 1 þ gð Þ

�2 vj j v2 þ b2

� �v2 � b2ð Þ1�g

, (11)

for jvj > b, and zero otherwise. The ESR intensity vanishesbelow a threshold set by the magnetic field, and falls off in apower law fashion, depending on the explicit value of Ks,and is shown in Fig. 1.

However, the sharp threshold disappears with increasingtemperature and the spectrum broadens. In the limit ofT � v, B and g � 1, which is relevant for realisticexperiments, the intensity can be approximated by (uponreinserting original units)

I vð Þ ¼ A2p �hvð Þ2

� h

�hv� KsgmBBð Þ2þh2þ h

�hvþ KsgmBBð Þ2þh2

" #,

(12)

where

h ¼ 2gpkBT : (13)

This expression works well outside of its range ofvalidity and it consists of two Lorentzians, centered around�KsgmBB, characterized by a width of h. Hence, theinteraction (g) together with the temperature determines

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Phys. Status Solidi B 246, No. 11–12 (2009) 2747

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0 0.5 1 1.5 20

10

20

30

40

50

60

70

80

90

100

gµBB/ ω

I(B)(arb.units)

K s = 1 .1

Figure 2 (online colour at: www.pss-b.com) The ESR signalintensity, Eq. (8) is shown as a function of the magnetic field forKs¼ 1.1 (corresponding to g ¼ 0.0045), kBT=�hv ¼ 0:1 (blue), 1(red), 5 (black), and 25 (magenta). The resonance occurs atgmBB=�hv ¼ 1=Ks 0:91 for small temperatures.

0 5 10 15 200

10

20

30

40

50

60

70

80

90

100

T/

I (K

sgµ

BB

=)(arb.units)

K s = 1. 1

K s = 1. 2

K s = 1. 3

K s = 1. 4

Figure 4 (online colour at: www.pss-b.com) The resonance peakin the ESR signal intensity is shown as a function of temperaturefor for Ks¼ 1.1, 1.2, 1.3, and 1.4. The resonance occurs atgmBB=�hv ¼ 1=Ks.

the width of the resonance and shifts the resonance center aswell, as is seen in Figs. 2 and 3 for Ks¼ 1.1 and 1.3,respectively. In Fig. 4, the evolution of the resonance peakwith temperature is shown for various Ks.

This expression allows us to make contact with theconventional Fermi liquid case. In that case,Ks¼ 1 (togetherwith g ! 0), which signals a non-interacting spin sector.The ESR intensity reduces to

I

1

1

2

2

3

3

4

4

5

I(B)(arb.units)

FiguinteKs¼(redgmB

www

vð Þ ¼ A2p2 gmBBð Þ2 d �hv� gmBBð Þ þ d �hvþ gmBBð Þ½ �:(14)

0 0.5 1 1.5 20

5

0

5

0

5

0

5

0

5

0

K s = 1 .3

gµBB/ ω

re 3 (online colour at: www.pss-b.com) The ESR signalnsity, Eq. (8) is shown as a function of the magnetic field for

1.3 (corresponding to g ¼ 0.0045), kBT=�hv ¼ 0:1 (blue), 1), 5 (black), and 25 (magenta). The resonance occurs atB=�hv ¼ 1=Ks 0:77 for small temperatures.

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Thus the integrated ESR intensity reads asRdvI vð Þ=v ¼ A4p2gmBB. In a Fermi liquid, this is

expressed in terms of the static spin-susceptibility [22], x0,as

RdvI vð Þ=v ¼ x0B

2?VpgmBB

�2m0�h. This fixes the so far

unknown numerical prefactor as A ¼ x0B2?V

�2m0�hp. In

summary, the ESR signal of an LL with broken spinrotational symmetry (i) is significantly broadened due tothe interaction and spin symmetry breaking and (ii) has asignal intensity which matches that of the non-interactingstate.

These results are similar to those found for the 1Dantiferromagnetic Heisenberg model [25], whose low energytheory is identical to the spin sector of a Luttinger liquid,Eq. (1). The ESR line-width also scales with T at lowtemperatures. However, the spin in the Heisenberg model,when represented in terms of fermionic variables via theJordan–Wigner transformation, usually contains non-localstring operators [21] and acquires a different scalingdimension than the spin of itinerant electrons. The exchangeanisotropy, causing the broadening of the ESR signal, sharescommon origin with the g-factor anisotropy in terms of SOcoupling, scaling with (Dg/g)2.

In Refs. [18, 19], a non-interacting spin sector wasconsidered (withKs¼ 1) together with explicitly SO coupledquasiparticles, leading to a narrow two peak spectrum. Ourapproach takes general spin anisotropies due to SO coupling,spin backscattering, dipole-dipole interactions and magneticfield into account in our starting Hamiltonian [Eq. (1)]. Theseprocesses introduce interactions in the spin sector [23],resulting in Ks 6¼ 1.

The SO coupling in SWCNTs was found to beunexpectedly large, around 1 meV for a nanotube withdiameter of 1 nm, resulting in a g-factor enhancement

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2748 B. Dora et al.: Single-wall carbon nanotubes: spintronics in the Luttinger liquid phasep

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g¼ 2.14 in a few electron carbon nanotube quantum dot [17].According to our theory, the position of the resonance occursat Ksg with the bare g-factor, which leads us to Ks� 1.07. Inaddition,Ks� 1.3 for quantum wires [26] like InAs, which isanother possible realization of LL. These materials possess aSO coupling of the same order of magnitude than SWCNTsbut they have a three times smaller Fermi velocity. Followinga similar line of reasoning for SWCNTs, we use [21]Ks 1 þ gs= 2pvFð Þ with gs the effective interaction in thespin sector using the g-ology notations [19, 21], influencedby the SO coupling (the same as for quantum wires [26], butwith vF 3vF;wire). Based on all this, we take a conservativeestimate of Ks to be around 1.1.

In the following, we discuss the relevance of the resultson the absence of an ESR signal in SWCNTs. As we showedabove, the ESR signal intensity of an LL crosses oversmoothly for the non-correlated case to the static suscepti-bility that is the Pauli susceptibility of metallic SWCNTs[27]: x0 ¼ m0g

2m2BD EFð Þ=4, where D eFð Þ is the density of

states (DOS) at the Fermi energy. To have an accurate valuefor the DOS in a realistic sample, we performed densityfunctional theory calculations with the Vienna ab initioSimulation Package [28] within the local density approxi-mation for metallic nanotubes. The projector augmented-wave method was used with a plane-wave cutoff energy of400 eV. The DOS was obtained with a Green’s functionapproach from the band structure. We considered theSWCNTs with chiral indices (9,9), (15,6), (10,10), (18,0),and (11,11) [29]. These tubes are within the Gaussiandiameter distribution of a usual SWCNT sample with a meandiameter of 1.4 nm and a variance of 0.1 nm. We confirmedby nearest-neighbor tight binding calculations on all themetallic SWCNTs in the above diameter distribution thatthe DOS depends very weakly on the chirality, thus the aboveSWCNTs chiralities are indeed representative for theensemble of the metallic tubes.

We obtain that such a tube ensemble has a DOS ofD EFð Þ ¼ 4:6 � 10�3 states/eV/atom by averaging the DOSfor the above SWCNTs and taking into account that only onethird of the tubes are metallic for this diameter range [29].This is a very low DOS which results from the one-dimensionality of the tubes and from the fact that themajority of the tubes are non-metallic. It is 50 times smallerthan in K3C60 [D(EF) 0.3 states/eV/atom [10]] and iscomparable to the low DOS of pristine graphite [D EFð Þ 5 � 10�3 states/eV/atom [30]]. With the above DOS, weobtain that a typical 2 mg SWCNT sample gives a practicallydetectable signal-to-noise ratio of S/N¼ 10 for a spectrummeasured for 1000 s provided the ESR line is not broaderthan 110 mT. To obtain this value, we considered that thestate-of-the-art ESR spectrometers give an S/N¼ 1 for 1010

S¼ 1/2 spins at 300 K provided the ESR line-width is 0.1 mTand each spectra points (typically 1000) are measured for 1 s.We also took into account that the S/N drops with the squareof the line-width for broadening beyond 1 mT.

The above calculated homogeneous broadening of theESR line of an LL is 2pgkBT=gmB in units of the magnetic

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field. Thus at 4 K, which is the lowest available temperaturefor most ESR spectrometers, one has a broadening of g18.7 Tesla. This, together with the above detectabilitycriterion gives an upper limit of g ¼ 6 � 10�3 for thedetection of the ESR signal. Clearly, the above conservativeestimate of g ¼ 4:5 � 10�3 based on the Ks¼ 1.1 value isclose to this limit, which explains why careful studies havenot yet yielded a conclusive ESR signal of itinerant electronsis SWCNTs. This argument might also be turned around: thefact that no ESR signal of the itinerant electron has beenobserved in the SWCNTs means that the line is broadenedbeyond observability, which is translated to g > 6 � 10�3,putting also Ks> 1.1. However, other factors, such as, e.g.,limited microwave penetration into the SWCNT samplefurther limits ESR experiments, leading to the unobserva-bility of ESR for smaller values of g and Ks as well.

We finally comment on the future viability of thisobservation. Clearly, ESR spectrometers operating to subKelvin temperatures are required. Observation of linearlytemperature dependent ESR line-width would be anunambiguous evidence for the observation of the ESR signalof itinerant electrons in the LL state. Such a temperaturedependence is fairly unusual as ESR line-width in metalsnormally tends to a residual value similar to the resistivity.

In summary, we extended the theory of ESR in an LL forthe case of broken spin-symmetry with interacting spin sector.We obtain a significant homogeneous broadening of the ESRline-width with increasing temperature, which explains theunobservability of ESR in SWCNTs and puts severeconstraints on the usability of SWCNTs for spintronics.

Acknowledgements The authors acknowledge usefuldiscussions with L. Forro and an illuminating exchange ofe-mails with A. De Martino. Supported by the Hungarian StateGrants (OTKA) F61733, K72613, NK60984, F68852, andK60576. B. D., V. Z. and F. S. acknowledge the Bolyaiprogramme of the HAS for support and the Marie Curie IEFProject NANOTRAN.

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